Linear generation theory of 2D and 3D periodic internal waves in a viscous stratified fluid

Abstract

We investigated analytically and experimentally 2D and 3D periodic internal waves generated by small harmonic oscillations of a plate and of an impermeable vertical cylindrical tube in an exponentially stratified viscous fluid. The linearized governing equations were solved by an integral transform method. The exact boundary conditions on the surface of a body, as well as the governing equations are satisfied if, in addition to propagating internal waves, internal boundary currents on the emitting surface are taken into account. On the basis of these two forms of fluid motion, we constructed a complete linear theory of the wave generation, without any external parameters. We calculated wave amplitudes and evolution along the beam of the so-called ‘St. Andrew's Cross’ wave shape, namely the number of maxima in the wave amplitude cross-section. The spatial decay of the wave was different in 2D and 3D problems due to geometry. The distance from the source, where transition from a bi-modal beam to the uni-modal beam takes place, is defined. Small viscosity smoothes out the singularity that arises in the wave field along the inviscid characteristics and in the critical angles. Experimental observations and probe measurements of a periodic wave pattern confirmed the theoretical results for the far field wave structure. The absolute values of calculated wave amplitudes differed from the experimental values by a factor less than 1.5. Indirect evidence of the internal boundary currents in Schlieren photographs of the flow pattern were presented. Copyright © 2001 John Wiley & Sons, Ltd.